STRUCTURE OF THE ELECTRON GAS AT THE SURFACE OF LIQUID HELIUM

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STRUCTURE OF THE ELECTRON GAS AT THE SURFACE OF LIQUID HELIUM

C. Grimes

To cite this version:

C. Grimes. STRUCTURE OF THE ELECTRON GAS AT THE SURFACE OF LIQUID HELIUM.

Journal de Physique Colloques, 1978, 39 (C6), pp.C6-1352-C6-1359. �10.1051/jphyscol:19786571�.

�jpa-00218062�

JOURNAL DE PHYSIQUE

Colloque C6, supplgment au no 8, Tome 39, aolit 1978, page ~ 6 - 1 3 5 2

STRUCTURE OF THE ELECTRON GAS AT THE SURFACE OF L I Q U I D HELIUM

C.C. Grimes

B e l l Telephone Laboratories, M m a y H i l l , New Jersey, U . S . A .

R&sum&.- On prdsente une breve revue de la structure du gaz 6lectronique 1 deux dimensions form6 par des electrons dans des &tats de surface induits par image-potentiel sur l'h6lium liquide. On fait ressortir le changement de la structure

1

mesure que la densit6 dlectronique par unit& de sur- face varie et que les propri6tds du syst~medvoluentdepuis celles d'un gaz dlectronique en passant par celles d'un liquide dlectronique, pour aboutir 1 celles d'un cristal 6lectronique.

Abstract.- A brief, tutorial survey is presented of the structure of the two-dimensional electron gas formed by electrons in image-potential induced surface states on liquid helium. Emphasis is placed on the change in structure as the electron areal density is varied and the system properties change from those of an electron gas to an electron liquid and finally to an electron crystal.

1 .

INTRODUCTION.- An electron in the vapor above a

liquid helium surface is attracted toward the sur- face by the classical image force. However, within a few Angstrijms of the surface the electron expe- riences a strong repulsive barrier to penetration into the helium. The repulsive barrier arises from the Pauli exclusion principle which requires that the wave function of an excess electron in the li- quid He must be orthogonal to the wave functions of the electrons on the He atoms. The potential well formed by the combination of the image potential and the repulsive barrier supports a series of bound electron states which are similar to the s states of the hydrogen atom, but bound with four orders of magnitude less energy. At

1

K and below such surface electrons are predominantly in the ground state within the image-potential well and are localized within 100 of the helium surface, but they are not localized in the plane of the sur- face and remain free to move parallel to the sur- face. A sheet of electrons on a liquid helium sur- face then forms a classical two-dimensional one- component Coulomb gas of variable density.

The purpose of this paper is to provide a brief tutorial review of what is known and what has been predicted concerning the structure of the two- dimensional electron gas on ahelium surface. The emphasis will be on the variation in properties as the electron areal density is increased at low tem- peratures and the behavior changes progressively from electron gas to liquid to solid.

The possibility that the potential well out- side a liquid helium surface could support a series of bound electron states was independently realized

by Sommer /I/ Cole and Cohen /2/ and by Shikin.131.

The first experimental investigations reported mea- surements of the mobility 141 and the trapping li- fetime

/ 5 / .

Subsequent experiments have included cyclotron resonance /6,7/ bound state spectroscopy

181

and a study of plasmon dispersion and damping 191. These experiments have confirmed the existence of surface state electrons (SSE), demonstrated that the effective mass is very nearly the free electron mass, measured the binding energy and Stark tuning rate, and measured the electron mobility in diffe- rent temperature regimes. At low temperatures (below 0.7 K) the SSE mobility parallel to the helium sur- face, which is limited by electron-ripplon scatte- ring, has a value near lo7 cm2/vs and decreases ap- proximately as 1/T for increasing temperature, T.

Above about 0.8 K the SSE mobility is limited by gas atom scattering and decreases nearly exponen- tially with increasing T.

The thermodynamic state of the 2D electron gas is determined by the ratio of the potential energy per electron to the kinetic energy. This dimensionless ratio, given by r

⁼

^rl/* n1I2 e 2 / k ~ where n is the areal density of electrons, charac- terizes the degree to which the electronic motions are correlated. At low ns and high temperature whe- re r is small, i.e. r < 1, the kinetic energy do- minates and the SSE behave as a 2D classical elec- tron gas. At higher densities and lower temperatu- res (1

5

r < 100) the electronic motions become more correlated and liquid like behavior is expec- ted. At still higher densities and low temperatures a 2D electron crystal is expected to form (for

r

2

100). Here we have assumed that for r

^<

^{100 the}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786571

SSE obey Maxwell-Boltzmann s t a t i s t i c s . This w i l l b e t r u e a s long a s kT >> EF where t h e Fermi energy f o r a 2D system i s given by EF = +fi2ns/m.

For s u r f a c e e l e c t r o n s on a l i q u i d t h e r e e x i s t s an upper bound on t h e a t t a i n a b l e e l e c t r o n a r e a l d e n s i t y . Gorkov and Chernikova / l o / pointed o u t t h a t t h e e x p r e s s i o n f o r t h e frequency of coupled plasmons and r i p p l o n s on a l i q u i d s u r f a c e c o n t a i n s competing terms t h a t can c a n c e l and cause t h e f r e - quency t o go t o z e r o f o r a f i n i t e wavelength a t a c r i t i c a l e l e c t r o n a r e a l d e n s i t y . The c r i t i c a l den- s i t y i s determined by t h e l i q u i d ' s s u r f a c e t e n s i o n and f o r 4 ~ e has t h e v a l u e N 2 2.2 x l o 9 A t t h i s a r e a l d e n s i t y t h e Fermi temperature i s o n l y

18 mK, s o t h e SSE on He obey Maxwell-Boltzman s t a - t i s t i c s a t a l l t h e temperatures s t u d i e d t h u s f a r (T > 0.3 K ) . For t h e remainder of t h i s paper we as- sume kT/EF >> 1 .

I n t h e remainder of t h i s paper we p r e s e n t i n s u c c e s s i o n b r i e f s e c t i o n s on t h e 2D e l e c t r o n g a s , l i q u i d , and c r y s t a l . Each s e c t i o n c o n t a i n s f i r s t a mention of t h e r e l e v a n t t h e o r e t i c a l works, then a

summary of t h e p e r t i n e n t experiments

-

^{i f any}

-

^that

have been performed, and f i n a l l y mention i s made of some obvious experiments remaining t o be done.

2. THE LOW DENSITY 2D ELECTRON GAS.- S e v e r a l theo- r e t i c a l papers have appeared r e c e n t l y which have t r e a t e d e x p l i c i t l y t h e c l a s s i c a l 2D e l e c t r o n gas.

A s mentioned above we expect g a s - l i k e behavior f o r

r

5 1 . This e x p e c t a t i o n i s borne o u t i n c a l c u l a t i o n s of t h e p a i r c o r r e l a t i o n f u n c t i o n by T o t s u j i / I l / a n d by Lado /12/ T o t s u j i performed numerical experiments by t h e Monte C a r l o method u s i n g 81 p a r t i c l e s w i t h i n a u n i t c e l l i n a p e r i o d i c system. He found t h a t f o r

r <

1 , t h e p a i r c o r r e l a t i o n f u n c t i o n i s a monotical-

l y i n c r e a s i n g f u n c t i o n of s e p a r a t i o n . O s c i l l a t o r y s t r u c t u r e i n t h e p a i r c o r r e l a t i o n f u n c t i o n , i n d i c a - t i n g t h e o n s e t of s h o r t range o r d e r , f i r s t appeared f o r 2.2 (

r

⁽ 2.7. Very s i m i l a r r e s u l t s were obtai- ned by Lado who solved n u m e r i c a l l y t h e hypernetted- chain i n t e g r a l e q u a t i o n . He a l s o found g a s - l i k e be- h a v i o r f o r s m a l l

r ,

b u t o s c i l l a t o r y s t r u c t u r e i n t h e p a i r c o r r e l a t i o n f u n c t i o n f i r s t appeared f o r 2.8 <

r

< 2.9 which i s a s l i g h t l y h i g h e r v a l u e t h a n found by T o t s u j i . Chalupa /13/ T o t s u j i /11,14/ and F e t t e r 1151 have c a l c u l a t e d t h e e q u a t i o n of s t a t e f o r t h e c l a s s i c a l 2D e l e c t r o n g a s . They f i n d t h a t t h e l e a d i n g c o r r e c t i o n s t o t h e i d e a l gas e q u a t i o n of s t a t e a r e of o r d e r

r2.

The electrodynamic and thermodynamic proper- t i e s t h e 2D c l a s s i c a l e l e c t r o n gas have been calcu- l a t e d by F e t t e r 1151 and by T o t s u j i 1141. They both employed many-body techniques t o c a l c u l a t e , i n t h e random phase approximation, t h e s c r e e n i n g of a s t a - t i c i m p u r i t y , t h e spectrum and damping of plasma o s c i l l a t i o n s , and i n e l a s t i c e l e c t r o n s c a t t e r i n g . The c a l c u l a t e d plasmon d i s p e r s i o n r e l a t i o n e x p l i c i t l y d i s p l a y s t h e e f f e c t s of Landau damping. They o b t a i - ned t h e e q u a t i o n of s t a t e and o t h e r thermodynamic f u n c t i o n s by ~ e b ~ e - H i i c k e l type expansions.

Few experiments have been performed i n t h e regime where l' .r_ 1 . Such experiments r e q u i r e small a r e a l d e n s i t i e s of e l e c t r o n s . For example : c o n s i d e r e l e c t r o n s on a l i q u i d He s u r f a c e a t T = 1 K, t h e n t h e c o n d i t i o n

r 5

l r e q u i r e s ns

5

l o 5 cm-2 which corresponds t o an e x t e r n a l l y a p p l i e d h o l d i n g e l e c - t r i c f i e l d of E ( 0 . 1 V/cm. I n a t y p i c a l expertmen- t a l c e l l which i s 1: 0 . 3 cm h i g h , t h e a p p l i e d poten- t i a l would be only 0.03 V. Thus t o m a n i p u l a t e such small s u r f a c e charge d e n s i t i e s r e q u i r e s c a r e t o avoid extraneous p o t e n t i a l o f f s e t s which can a r i s e from thermal EMFs and charge i n a d v e r t e n t l y d e p o s i t e d on t h e He f i l m which c o a t s a l l s u r f a c e s i n t h e ex- p e r i m e n t a l c e l l . Measurements employing s m a l l sur- f a c e charge d e n s i t i e s , a l s o , of c o u r s , y i e l d r e l a - t i v e l y small s i g n a l s .

For t h e r e a s o n s mentioned above, most expe- r i m e n t s have been performed i n t h e regime

10 5

r 5

100. However, Bridges and McGill have per- formed a time-of-flight m o b i l i t y measurement u s i n g small a r e a l d e n s i t i e s of e l e c t r o n s on He 1161. They employed p u l s e s of e l e c t r o n s c o n s i s t i n g of about 30 t o 600 e l e c t r o n s p e r p u l s e which occupied an a r e a of 0.2 cm2 r e s u l t i n g i n n < l o 4 cm-*. T h e i r lowest temperature was about 1.2 K y i e l d i n g

r

5 0.2.

A t 1.2 K t h e i r e l e c t r o n m o b i l i t y f e l l s i g n i f i c a n t l y below t h e m o b i l i t y obtained by o t h e r s a t t h e same temperature b u t w i t h d e n s i t i e s i n t h e e l e c t r o n li- quid regime /4,6,17/ Platzman e t a l . 1181 have c a l - c u l a t e d t h e d e n s i t y and frequency dependence of t h e c o n d u c t i v i t y when r i p p l o n s c a t t e r i n g i s dominant.

They f i n d t h a t t h e e l e c t r o n m o b i l i t y should b e an i n c r e a s i n g f u n c t i o n of d e n s i t y due t o s c r e e n i n g of t h e r i p p l o n s by t h e e l e c t r o n s . However, due t o a f i n i t e frequency approximation t h a t they employed, t h e i r c a l c u l a t i o n i s n o t d i r e c t l y a p p l i c a b l e t o t h e Bridges and McGill experiment. F u r t h e r work i s nee- ded t o e s t a b l i s h i f t h e observed m o b i l i t y i s a n

" e l e c t r o n gas" e f f e c t .

C6-1354

JOURNAL DE PHYSIQUE

There a r e a number of i n t e r e s t i n g experiments remaining t o be performed on t h e c l a s s i c a l 2D e l e c - t r o n g a s . O b s e r v a t i o n of t h e e f f e c t s of Landau dam- p i n g on plasmon d i s p e r s i o n and damping would b e of c o n s i d e r a b l e i n t e r e s t . I n such a n experiment one would b e of c o n s i d e r a b l e i n t e r e s t . I n such an expe- r i m e n t one would e x p e c t t o s e e t h e p r e d i c t e d t r a n s i - t i o n from t h e plasmon mode (w a k1I2) t o a sound-

P

l i k e c o l l e c t i v e mode a t low a r e a l d e n s i t i e s 115,191.

Any experiment, such a s l i g h t s c a t t e r i n g , t h a t could y i e l d i n f o r m a t i o n on t h e s t r u c t u r e f a c t o r would b e of g r e a t i n t e r e s t . F e t t e r h a s p o i n t e d o u t t h a t t h e 2D e l e c t r o n gas may be more e a s i l y s t u d i e d by u s i n g N e a s a s u b s t r a t e i n s t e a d of He 1151. By working w i t h s o l i d Ne n e a r T = l o 0 , much h i g h e r a r e a l den- s i t i e s (n

<

lo7) could b e used and s t i l l have

r

^<, 1.

3 . THE CLASSICAL 2D ELECTRON LIQUID.- I n t h e e l e c - t r o n l i q u i d regime 1

< r <

100, no s m a l l e x p a n s i o n parameter e x i s t s s o c a l c u l a t i o n s a r e d i f f i c u l t . Re- c e n t l y , however, t h e r e have appeared t h r e e computer c a l c u l a t i o n s on t h e s t r u c t u r e of t h e 2D e l e c t r o n li- q u i d : T o t s u j i ' s Monte C a r l o c a l c u l a t i o n

/

1 1 1 which t r e a t e d t h e parameter r a n g e 0.16 (

r

( 5 0 , Lado's h y p e r n e t t e d c h a i n c a l c u l a t i o n 1121 which spanned 0.1 (

r

( 100, and t h e m o l e c u l a r dynamics c a l c u l a - t i o n by Hockney and Brown 1201 which spanned t h e r a n g e 33 (

r

( 1850. As mentioned above, t h e p a i r c o r r e l a t i o n f u n c t i o n s t a r t s t o e x h i b i t o s c i l l a t o r y s t r u c t u r e a s

r

exceeds t h e v a l u e 2.7.

A s

r

i n c r e a s e s , t h e s h o r t r a n g e o r d e r i n c r e a - s e s and t h e a m p l i t u d e of t h e o s c i l l a t o r y s t r u c t u r e i n t h e p a i r c o r r e l a t i o n f u n c t i o n i n c r e a s e s . The gra- d u a l growth i n s h o r t r a n g e o r d e r w i t h i n c r e a s i n g

r

i s i l l u s t r a t e d by t h e schematic p a i r c o r r e l a t i o n f u n c t i o n s shown i n f i g u r e 1. The s t r u c t u r e f a c t o r d i s p l a y s a s i m i l a r growth i n o s c i l l a t o r y s t r u c t u r e w i t h i n c r e a s i n g T. The p h y s i c a l p i c t u r e suggested by t h e c a l c u l a t i o n s i s one i n which i t i s becoming i n c r e a s i n g l y p r o b a b l e t h a t a g i v e n e l e c t r o n w i l l b e surrounded by s i x n e a r e s t n e i g h b o r s each l y i n g c l o s e t o t h e s i t e s of a t r i a n g u l a r l a t t i c e , b u t w i t h o u t having any long-range c o r r e l a t i o n i n p o s i t i o n s . At

l a r g e v a l u e s of

r

( n e a r r-100) one can v i s u a l i z e each e l e c t r o n a s o s c i l l a t i n g r a p i d l y i n a p o t e n t i a l w e l l due t o i t s n e i g h b o r s a s i t wanders s l o w l y from one c o n f i g u r a t i o n t o a n o t h e r .

Although most of t h e e x p e r i m e n t s on SSE on l i q u i d He have been performed i n t h e e l e c t r o n l i q u i d regime, o n l y one experiment, t h a t of Z i p f e l , Brown

0 2 4 6

- _a r

Fig. 1 : S k e t c h of t h e p a i r c o r r e l a t i o n f u n c t i o n f o r a c l a s s i c a l two-dimensional e l e c t r o n g a s f o r v a l u e s of t h e plasma parameter a s l a b e l e d

and Grimes (ZBG) h a s y i e l d e d d i r e c t i n f o r m a t i o n p e r t a i n i n g t o t h e s h o r t r a n g e o r d e r of t h e e l e c t r o n l i q u i d 1211. That experiment was a n e x t e n s i o n of a s t u d y mentioned i n t h e i n t r o d u c t i o n r e g a r d i n g t h e s p e c t r a of t r a n s i t i o n s between bound e l e c t r o n i c s t a - t e s i n t h e image p o t e n t i a l - w e l l 181. The e a r l i e r ex- p e r i m e n t s measured t h e s p l i t t i n g s between t h e ground s t a t e and t h e f i r s t two e x c i t e d s t a t e s by o b s e r v i n g nun wave a b s o r p t i o n a s t h e s p l i t t i n g s were S t a r k - tuned i n t o r e s o n a n c e w i t h t h e f r e q u e n c y of t h e ap- p l i e d mm r a d i a t i o n . By a p p l y i n g a magnetic f i e l d B

!!

p a r a l l e l t o t h e l i q u i d He s u r f a c e , ZBG o b t a i n e d Sfom t h e s p e c t r o s c o p y experiment a t i m e , T ~ t h a t , i s ex- p e c t e d t o b e c l o s e l y r e l a t e d t o t h e e l e c t r o n v e l o c i - t y a u t o - c o r r e l a t i o n time. Applying BI1 c o u p l e s t h e s p e c t r o s c o p i c t r a n s i t i o n s t o t h e component of t h e e l e c t r o n i c motion p a r a l l e l t o t h e He s u r f a c e b u t p e r p e n d i c u l a r t o t h e m a g n e t i c f i e l d . P h y s i c a l l y , t h i s c o u p l i n g i s due t o t h e L o r e n t z f o r c e which i s e q u i v a l e n t t o an e l e c t r i c f % e l d normal t o t h e He s u r f a c e and t h u s a c t s a s a p e r t u r b a t i o n t o t h e Stark t u n i n g f i e l d . S i n c e t h i s p e r t u r b a t i o n i s p r o p o r t i o - n a l t o a component f o t h e e l e c t r o n s v e l o c i t y , we e x p e c t t h a t a t low SSE a r e a l d e n s i t i e s a p p l i c a t i o n of B1/ w i l l simply broaden t h e s p e c t r o s c o p i c l i n e - shape i n t o a Gaussian whose w i d t h i s a measure of t h e t e m p e r a t u r e of t h e e l e c t r o n g a s . However, a t h i g h a r e a l d e n s i t i e s t h e e l e c t r o n - e l e c t r o n s c a t t e - r i n g f r e q u e n c y becomes g r e a t e r and t h e c o n t r i b u t i o n

t o t h e l i n e w i d t h due t o B l l i s a p p r e c i a b l y reducedby motional averaging. I n t h e m o t i o n a l l y narrowed r e g i - me t h e l i n e w i d t h c o n t a i n s a c o n t r i b u t i o n t h a t i s p r o p o r t i o n a l t o T

.

Values of T~ a t d i f f e r e n t a r e a l d e n s i t i e s were deduced from t h e observed linewidths, and were found t o d e c r e a s e from 9.9 x 1 0 - l ~ s t o

1.4 x 10-l1 s a s

r

i n c r e a s e d from 9 t o 36. ZBG made t h e p h y s i c a l l y p l a u s i b l e a s s e r t i o n t h a t T i s t h e e l e c t r o n v e l o c i t y a u t o - c o r r e l a t i o n time. They a l s o p o i n t e d o u t t h a t t h e observed v a l u e s of r a r e c l o - s e t o t h e p e r i o d s expected f o r harmonic o s c i l l a t i o n of a n e l e c t r o n about a l a t t i c e s i t e i n a t r i a n g u l a r e l e c t r o n l a t t i c e of t h e same a r e a l d e n s i t y .

I t o h e t a l . d e r i v e d f o r t h e 2D e l e c t r o n li- quid an e x p r e s s i o n t h a t r e l a t e d t h e v e l o c i t y auto- c o r r e l a t i o n time t o t h e r a d i a l d i s t r i b u t i o n func- t i o n 1221. Using t h e p a i r c o r r e l a t i o n f u n c t i o n s from t h e computer c a l c u l a t i o n s of Hockney and Brown 1201 and of T o t s u j i / 1 1 / t h e y t h e n c a l c u l a t e d t h e v e l o c i t y a u t o - c o r r e l a t i o n times a t s e v e r a l v a l u e s of T. The v e l o c i t y . a u t o - c o r r e l a t i o n times t h u s ob- t a i n e d agreed w i t h t h e v a l u e s of r measured by ZBG w i t h i n t h e experimental e r r o r of

+

10 %. I t o h e t a l . a l s o used a harmonic l a t t i c e model t o c o n s t r u c t an approximate r a d i a l d i s t r i b u t i o n f u n c t i o n which was then employed t o c a l c u l a t e a u t o - c o r r e l a t i o n times.

The a u t o - c o r r e l a t i o n times t h u s o b t a i n e d were about 30 % l a r g e r t h a n t h e measured v a l u e s of rc. They conclude from t h e c l o s e correspondence of t h e s e numbers t h a t t h e s h o r t range o r d e r i n t h e e l e c t r o n

l i q u i d even a t

r

= 36 i s much l i k e t h a t i n t h e e l e c t r o n c r y s t a l .

Platzman and H a l p e r i n (PH) have examined i n d e t a i l t h e q u a n t i t i e s t h a t can be measured i n t h e ZBG t y p e of experiment /23/. They f i n d t h a t t h e in- t e r p r e t a t i o n of t h e experiment i s complicated i n a l l b u t a few l i m i t i n g c a s e s . Omitting r i p p l o n and vapor atom s c a t t e r i n g , they f i n d t h e i n t e r p r e t a t i o n i s simple i n t h e low d e n s i t y l i m i t

r

^<< 1 . I n t h i s c a s e t h e approach taken by ZBG i s borne o u t : t h e broadening of t h e l i n e s h a p e by t h e applied. f i e l d i s p a r t i a l l y removed by motional narrowing l e a d i n g t o a l i n e w i d t h p r o p o r t i o n a l t o r t h e v e l o c i t y auto-

a c ' c o r r e l a t i o n time.

A t ' h i g h e r d e n s i t i e s i n t h e e l e c t r o n l i q u i d regime, t h e p e l o c i t y a u t o - c o r r e l a t i o n f u n c t i o n i s expected t o develop o s c i l l a t o r y s t r u c t u r e j u s t a s t h e p a i r c o r r e l a t i o n f u n c t i o n does. (Hansen, McDonald and P o l l o c k / 2 4 / have c a l c u l a t e d v e l o c i t y a u t o - c o r r e l a t i o n f u n c t i o n s f o r t h e

three

dimensional

e l e c t r o n l i q u i d and f i n d t h a t a s

r

i n c r e a s e s t h e os- c i l l a t o r y c o n t r i b u t i o n from t h e plasmon i n c r e a s e s ) . The PH a n a l y s i s t h e n i n d i c a t e s t h a t t h e a b s o r p t i o n l i n e s h a p e should no longer b e simple, b u t i s r e l a - t e d t o t h e F o u r i e r t r a n s f o r m of t h e v e l o c i t y auto- c o r r e l a t i o n f u n c t i o n and t h e r e f o r e c o n t a i n s s t r u c - t u r e . The i n t e r p r e t a t i o n of experimental l i n e s h a p e s should t h e n be more complicated than t h e procedure used by ZBG. However, i n view of I t o h e t a l ' s . f i n - ding of c l o s e agreement between t h e ZBG -rc and t h e T c a l c u l a t e d from t h e p a i r c o r r e l a t i o n f u n c t i o n s ,

a c

i t appears t h a t r c i s a c t u a l l y a measure of t h e ve- l o c i t y a u t o - c o r r e l a t i o n t i m e / 2 2 / . The e x t r a s t r u c t u - r e i n t h e l i n e s h a p e p r e d i c t e d by PH was n o t s e e n i n t h e experimental t r a c e s .

Platzman and H a l p e r i n emphasize t h a t t h e ZBG type of experiment, i f performed a t lower temperatu- r e s and i n a l a r g e r r a n g e of magnetic f i e l d s , could y i e l d i n f o r m a t i o n on t h e dynamical p r o p e r t i e s of t h e e l e c t r o n l i q u i d o r e l e c t r o n c r y s t a l . The l i n e - shape should c o n t a i n i n f o r m a t i o n on s e l f - d i f f u s i o n i n t h e l i q u i d and on t r a n s v e r s e modes, d e f e c t motion and zone boundary phonons i n t h e s o l i d .

The e f f e c t s of e l e c t o n - r i p p l o n a n electron-He atom s c a t t e r i n g on s p e c t r o s c o p i c l i n e w i d t h s b o t h w i t h and w i t h o u t a n a p p l i e d magnetic f i e l d have been analyzed by Ando 1251. His r e s u l t s , which a r e a p p l i - c a b l e i n t h e low-density, 'l < 1 , regime enphasize t h a t t h e l i n e w i d t h c o n t a i n s c o n t r i b u t i o n s from broa- dening of b o t h t h e ground s t a t e and t h e e x c i t e d s t a t e . H i s c a l c u l a t i o n s should prove v e r y u s e f u l when t h e spectroscopy expercments a r e extended t o

lower temperatures and lower e l e c t r o n d e n s i t i e s . Many o t h e r experiments performed i n t h e l i q u i d regime have not y e t y i e l d e d i n f o r m a t i o n on t h e s t r u c - t u r e of t h e l i q u i d . For example : c y c l o t r o n resonan- ce and m o b i l i t y measurements have y?elded only "one e l e c t r o n " p r o p e r t i e s of t h e system. Experimental s t u d i e s of t h e l o s g wavelength plasmon d i s p e r s i o n , although t h e y probe a c o l l e c t i v e mode of t h e system, a r e n o t s e n s i t i v e t o t h e short-range s t r u c t u r e . I n f a c t , t h e long wavelength plasmon d i s p e r s i o n r e l a - t i o n remains unchanged throughout t h e l i q u i d and c r y s t a l regimes. Of c o u r s e , i f t h e plasmon s t u d i e s could be extended t o s h o r t wavelengths n e a r t h e B r i l l o u i n zone boundary, t h e n t h e plasmon d i s p e r - s i o n becomes s e n s i t i v e t o t h e s h o r t range s t r u c t u r e / 1 9 / . Spectroscopy w i t h o u t a magnetic f i e l d a t mm and sub-mm wavelengths employs l a r g e S t a r k tuning e l e c t r i c f i e l d s and p e r m i t s u s e of high s u r f a c e

JOURNAL DE PHYSIQUE

charge d e n s i t i e s . A t high charge d e n s i t i e s

(r *

36) t h e s p e c t r o s c o p i c l i n e develops s i d e bands t h a t mo- v e o u t w i t h i n c r e a s i n g n 1261. These sidebands may be due t o "zone boundary" plasmons i n t h e h i g h l y c o r r e l a t e d l i q u i d 1271.

C l e a r l y , t h e r e a r e many i n t e r e s t i n g e x p e r i - ments remaining t o b e performed on t h e e l e c t r o n li- quid. The spectroscopy-in-a-magnetic-field (ZBG) type of experiment i s v e r y promising and w a r r a n t s a d d i t i o n a l work. Again, any experiment t h a t could measure t h e s t r u c t u r e f a c t o r would be v e r y u s e f u l .

4. THE HIGH DENSITY 2D COULOMB CRYSTAL.- A t high e l e c t r o n a r e a l d e n s i t i e s ,

r

5 100, t h e Coulomb po- t e n t i a l energy predominates and t h e s t a t e of lowest energy becomes t h e t r i a n g u l a r e l e c t r o n l a t t i c e . T h i s 2D analogue of Wigner c r y s t a l l i z a t i o n was f i r s t d i s c u s s e d by Grandall and Williams 1281 w h i l e t h e phase diagram was f i r s t c a l c u l a t e d by Platzman and Fukuyama 1291. The Hockney and Brown (HB) molecular dynamics c a l c u l a t i o n p l a c e s t h e m e l t i n g t r a n s i t i o n a t

r

⁼ 95 1201. The phase diagram based on t h a t va- l u e i s sketched i n f i g u r e 2. One of t h e most i n t e - r e s t i n g r e s u l t of t h e HB computer experiment i s t h e d i s c o v e r y of t h e lambda-type anomaly i n t h e e l e c t r o - n i c s p e c i f i c h e a t a t t h e l i q u i d - s o l i d phase t r a n s i - t i o n . They f i n d t h a t a l l o t h e r c a l c u l a t e d p r o p e r t i e s ( i n t e r n a l energy, p a i r c o r r e l a t i o n f u n c t i o n , s t r u c - t u r e f a c t o r ) v a r y smoothly through t h e phase t r a n - s i t i o n , so it appears t o b e a continuous liquid-so- l i d phase t r a n s i t i o n .

To d a t e , no f i r m l a b o r a t o r y experimental e v i - d e n c e f o r t h e e x i s t e n c e of t h e 2D e l e c t r o n c r y s t a l

h a s appeared. N e v e r t h e l e s s , t h e concept of e l e c t r o n c r y s t a l l i z a t i o n h a s s t i m u l a t e d many persons t o w r i t e t h e o r e t i c a l p a p e r s . A p a r t i a l l i s t i n g f o l l o w s :

The phonon s p e c t r a of t r i a n g u l a r /29,30/ and s q u a r e 130-331 e l e c t r o n l a t t i c e s have been c a l c u l a - t e d , and t h e square l a t t i c e been shown t o b e dyna- m i c a l l y u n s t a b l e 130,331. The l o n g i t u d i n a l phonon mode of t h e e l e c t r o n c r y s t a l i s j u s t t h e 2D plasmon

w 2 = 2nnse2 k/m which i s d i s p e r s i v e , whereas t h e

t r a n s v e r s e mode wt = ck i s sound-like a t long wave- l e n g t h s . A magnetic f i e l d a p p l i e d normal t o t h e He s u r f a c e couples t h e two phonon modes /30,32,34,35/.

I n a s t r o n g e l e c t r i c f i e l d EL a dimple can form i n t h e He s u r f a c e under each e l e c t r o n and form a har- monic o s c i l l a t o r p o t e n t i a l w e l l . The d i s p e r s i o n ' r e -

l a t i o n s f o r t h e coupled harmonic o s c i l l a t o r and pho- non mades have been d e r i v e d by S h i k i n 1361.

Fig. 2 : Phase diagram f o r a s h e e t of e l e c t r o n s on t h e surf,ace of l i q u i d He. N denotes t h e e l e c t r o n a r e a l d e n s i t y i n ~ m - ~ . The s o l i d - l i q u i d phase boun- d a r y i s taken t o occur where t h e plasma parameter,

r ,

has t h e v a l u e 95 I201

S h i k i n has proposed a novel experiment t o d e t e c t t h e presence of t h e e l e c t r o n c r y s t a l on l i q u i d He /37/.He suggested u s i n g t h e p e r i o d i c e l e c t r o n lat- t i c e a s a t r a n s d u c e r t o e x c i t e s t a n d i n g c a p i l l a r y waves on t h e Hu s u r f a c e . Driving t h e e l e c t r o n c r y s t a l w i t h an r f f i e l d a p p l i e d normal t o t h e s u r f a c e should e x c i t e resonances when t h e frequency and t h e low index r e c i p r o c a l l a t t i c e v e c t o r s q i j of t h e e l e c t r o n l a t t i c e s a t i s f y t h e c a p i l l a r y wave d i s p e r s i o n r e l a - t i o n w2 = ( a l p )

q3.

where a i s t h e s u r f a c e t e n s i o n

1.l

c o e f f i c i e n t of He and p i s t h e d e n s i t y . TEis propo- sed experiment was analyzed i n d e t a i l by Monarkhaand S h i k i n 1381. I f observed, t h e resonances would pro- v i d e c l e a r evidence f o r t h e e x i s t e n c e of t h e e l e c - t r o n c r y s t a l .

5. CONCLUSION.- It i s c l e a r t h a t much more theore- t i c a l work h a s been done on t h e 2D e l e c t r o n system t h a n experimental work. There i s an e v i d e n t n e e d f o r experimental t e s t s of- t h e t h e o r e t i c a l models and p r e d i c t i o n s .

I f t h e r e a d e r h a s been i n t r i g u e d by t h i s sim- p l e , b u t many f a c e t e d system, and would l i k e t o r e a d f u r t h e r , an u n c r i t i c a l review w i t h a f a i r l y complete l i s t of r e f e r e n c e s appeared r e c e n t l y 1391. E a r l i e r reviews were w r i t t e n by S h i k i n and Monarkha 1401 C r a n d a l l 1411 and Cole 1421.

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JOURNAL DE PHYSIQUE

POST DEADLINE PAPERS ACCEPTED FOR CONTRIBUTION

-

^-t

- Magnetic fields effects on the helical instability of the

fil

I V textures in 3 ~ e - ~ . W.M. Saslow, C.R. Hu

- Helical textures in 3 ~ e - ~ in the presence of superflow and magnetic field .

Y.R. Lin-Liu, K. Maki, D. Vollhardt

- Positive ion structure in a tricritical 3 ~ e - 4 ~ e mixture .

M. Wanner, P. Leiderer

- Evidence for a new high field phase transition in solid 3 ~ e .

E.A. Schuberth, D.M. Bakalyar, E.D. Adams

- Neutron inelastic scattering from liquid 3 ~ e at 40

mK

and at 1.2 K .

K. ~kSld, G.A. Pelizzari

- Measurement of the lifetime of 4 ~ e rotons

by

neutron spin echo .

F. Mezei

- Adsorption of the first layer

of

argon on graphite .

F. Millot